Problem: Emma had just been given some coins by her parents.  On the way to school she lost exactly half of them, and then by retracing her steps she found exactly four-fifths of the coins she had lost.  What fraction of the coins that she received from her parents were still missing after Emma retraced her steps? Express your answer as a common fraction.
Answer: Let the number of coins Emma received from her parents be $x$. She lost $\frac{1}{2}x$ coins on the way to school. She found $\frac{1}{2}\cdot\frac{4}{5}x=\frac{2}{5}x$ of the coins by retracing her steps. Thus, Emma has $\frac{1}{2}x + \frac{2}{5}x=\frac{9}{10}x$ coins. She is still missing $x-\frac{9}{10}x=\frac{1}{10}x$ coins, so she is missing $\boxed{\frac{1}{10}}$ of the coins.